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Solvability of Linear Differential Systems in the Liouvillian Sense

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Solvability of Linear Differential Systems in the Liouvillian Sense Solvability of linear differential systems in the Liouvillian sense R. R. Gontsov, I. V. Vyugin∗ Abstract The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular singu- lar points as well as those with (non-resonant) irregular ones and propose some criteria of solvability for systems whose (formal) exponents are sufficiently small. 1 Introduction Consider on the Riemann sphere C a linear differential system dy = B(z) y, y(z) ∈ Cp, (1) dz of p equations with a meromorphic coefficient matrix B(z) having singularities at points a1, . , an. A singular point z = ai is said to be regular, if any solution of the system has at most polyno- mial growth in any sector of small radius with vertex at this point and an opening less than 2π. Otherwise the point z = ai is said to be irregular. The Picard–Vessiot extension of the field C(z) of rational functions corresponding to the system (1) is a differential field F obtained by adjoining to C(z) all entries of a fundamental matrix Y (z) of the system (1). One says that the system (1) is solvable by quadratures, if the entries of the matrix Y (z) are expressed in elementary or algebraic functions and their integrals or, more formally, if the field F is contained in some extension of C(z) obtained by adjoining algebraic functions, exponentials or integrals: C(z)= F1 ⊆ . ⊆ Fm, F ⊆ Fm, where Fi = Fihxii (i = 1,...,m − 1), and either xi is algebraic over Fi, or xi is an exponential arXiv:1312.2518v1 [math.CA] 9 Dec 2013 +1 of an element in Fi, or xi is an integral of an element in Fi. Such an extension C(z) ⊆ Fm is called Liouvillian, thus solvability by quadratures means that the Picard–Vessiot extension F is contained in some Liouvillian extension of the field of rational functions. Solvability or non-solvability of a linear differential system by quadratures is related to properties of its Galois group. The differential Galois group G = Gal (F/C(z)) of the system (1) (of the Picard–Vessiot extension C(z) ⊆ F ) is the group of differential automorphisms of the field F (i. e., automorphisms commuting with differentiation) that preserve elements of the field C(z): d d G = σ : F → F σ ◦ = ◦ σ, σ(f)= f ∀f ∈ C(z) . n dz dz o ∗The author is partially supported by Dynasty Foundation, Simons-IUM fellowship and grant RFBR 12-01- 33058. 1 As follows from the definition, the image σ(Y ) of the fundamental matrix Y (z) of the system (1) under any element σ of the Galois group is a fundamental matrix of this system again, that is, σ(Y )= Y (z)C, C ∈ GL(p, C). As every element of the Galois group is determined uniquely by its action on a fundamental matrix of the system, the Galois group G can be regarded as a subgroup of the matrix group GL(p, C). Moreover this subgroup G ⊂ GL(p, C) is algebraic, i. e., closed in the Zariski topology of the space GL(p, C) (the topology whose closed sets are those determined by systems of polynomial equations), see [12, Th. 5.5]. The Galois group G can be represented as a union of finite number of disjoint connected sets that are open and closed simultaneously (in the Zariski topology), and the set containing the identity matrix is called the identity component. The identity component G0 ⊂ G is a normal subgroup of finite index [12, Lemma 4.5]. Due to the Picard–Vessiot theorem, solvability of the system (1) by quadratures is equivalent to solvability of the subgroup G0 [12, Th. 5.12], [13, Ch. 3, Th. 5.1]. (Recall that a group H is said to be solvable, if there exist intermediate normal subgroups {e} = H0 ⊂ H1 ⊂ . ⊂ Hm = H such that each factor group Hi/Hi−1 is Abelian, i = 1,...,m.) Alongside the Galois group one considers the monodromy group M of the system (1) gen- erated by the monodromy matrices M1,...,Mn corresponding to analytic continuation of a fundamental matrix Y (z) around the singular points a1, . , an. (The matrix Y (z) considered in a neighbourhood of a non-singular point z0 goes to Y (z)Mi under an analytic continuation along a simple loop γi encircled a point ai.) As the operation of analytic continuation commutes with differentiation and preserves elements of the field C(z) (single-valued functions), one has M ⊆ G. Furthermore the Galois group of a system whose singular points are all regular is a closure of its monodromy group (in the Zariski topology, see [13, Ch. 6, Cor. 1.3]), hence such a system is solvable by quadratures, if and only if the identity component of its monodromy group is solvable. We are interested in the cases when the answer to the question concerning solvability of a linear differential system by quadratures can be given in terms of the coefficient matrix of the system. For example, in the case of a Fuchsian system (a particular case of a system with regular singular points) n dy Bi = y, Bi ∈ Mat(p, C), (2) dz z − ai Xi=1 whose coefficients Bi are sufficiently small, Yu. S. Ilyashenko and A. G. Khovansky [9] have ob- tained an explicit criterion of solvability. Namely, the following statement holds: There exists ε = ε(n,p) > 0 such that a condition of solvability by quadratures for the Fuchsian system (2) with kBik < ε takes an explicit form: the system is solvable by quadratures, if and only if all the matrices Bi are triangular (in some basis). In this article we propose a refinement of the above assertion in which it is sufficient that the eigenvalues of the residue matrices Bi be small (the estimate is given), and we also propose a generalization to the case of a system with irregular singular points. 2 A local form of solutions of a system near its singular points A singular point ai of the system (1) is said to be Fuchsian, if the coefficient matrix B(z) has a simple pole at this point. Due to Sauvage’s theorem, a Fuchsian singular point of a linear differential system is regular (see [8, Th. 11.1]). However, the coefficient matrix of a system at a regular singular point may 2 in general have a pole of order greater than one. Let us write the Laurent expansion of the coefficient matrix B(z) of the system (1) near its singular point z = a in the form B−r−1 B−1 B(z)= + . + + B + ..., B− − 6= 0. (3) (z − a)r+1 z − a 0 r 1 The number r is called the Poincar´erank of the system (1) at this point (or the Poincar´erank of the singular point z = a). For example, the Poincar´erank of a Fuchsian singularity is equal to zero. The system (1) is said to be Fuchsian, if all its singular points are Fuchsian (then it can be written in the form (2)). The system whose singular points are all regular will be called regular singular. According to Levelt’s theorem [15], in a neighbourhood of each regular singular point ai of the system (1) there exists a fundamental matrix of the form e Ai Ei Yi(z)= Ui(z)(z − ai) (z − ai) , (4) 1 p where Ui(z) is a holomorphic matrix at the point ai, Ai = diag(ϕi ,...,ϕi ) is a diagonal integer j matrix whose entries ϕi organize in a non-increasing sequence, Ei = (1/2πi) ln Mi is an upper- triangular matrix (the normalized logarithm of the corresponding monodromy matrix) whose j e f eigenvalues ρi satisfy the condition j 0 6 Re ρi < 1. Such a fundamental matrix is called a Levelt matrix, and one also says that its columns form a Levelt basis in the solution space of the system (in a neighborhood of the regular singular point j j j ai). The complex numbers βi = ϕi + ρi are called the (Levelt) exponents of the system at the regular singular point ai. If the singular point ai is Fuchsian, then the corresponding matrix Ui(z) in the decomposition (4) is holomorphically invertible at this point, that is, det Ui(ai) 6= 0. It is not difficult to check that in this case the exponents of the system at the point ai coincide with the eigenvalues of the residue matrix Bi. And in the general case of a regular singularity ai there are estimates for the order of the function det Ui(z) at this point obtained by E. Corel [4] (see also [6]): p(p − 1) r 6 ord det U (z) 6 r , i ai i 2 i where ri is the Poincar´erank of the regular singular point ai. These estimates imply the inequalities for the sum of exponents of the regular system over all its singular points, which are called the Fuchs inequalities: n n p n p(p − 1) − r 6 βj 6 − r (5) 2 i i i Xi=1 Xi=1 Xj=1 Xi=1 (the sum of exponents is an integer). Let us now describe the structure of solutions of the system (1) near one of its irregular singular points. We suppose that the irregular singularity ai of Poincar´erank ri is non-resonant, 1 p that is, the eigenvalues bi , . , bi of the leading term B−ri−1 of the matrix B(z) in the expansion (3) at this point are pairwise distinct.
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